On functional equations related to derivations in semiprime rings and standard operator algebras

Abstract

Abstract. In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra. Suppose there exist linear mappingsThroughout, R will represent an associative ring with center Z(R). As usual we write [x, y] for xy − yx. Given an integer n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime, if for a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. We denote by Q s the symmetric Martindale ring of quotients. For the explanation of Q s we refer the reader to [2]. Let A be an algebra over the real or complex field and let B be a subalgebra of A.

Hyers–Ulam Stability of an Additive-Quadratic Functional Equation

“…Hyers theorem was generalized by Aoki [3] for additive mappings and Rassias [12] for quadratic mappings. During the last three decades the stability theorem of Rassias [26] provided a lot of influence for the development of stability theory of a large variety of functional equations (see [1,2,4,7,9,11,14,17,18,21,22,23,27]). One of the most famous functional equations is the following additive functional equation g(x + y) = g(x) + g(y)…”